Infinite stable looptrees

Abstract

We give a construction of an infinite stable looptree, which we denote by ${\mathcal{L}}_{\alpha }^{\infty }$, and prove that it arises both as a local limit of the compact stable looptrees of Curien and Kortchemski (2015), and as a scaling limit of the infinite discrete looptrees of Richier (2017) and Björnberg and Stefánsson (2015). As a consequence, we are able to prove various convergence results for volumes of small balls in compact stable looptrees, explored more deeply in a companion paper. We also establish the spectral dimension of ${\mathcal{L}}_{\alpha }^{\infty }$, and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on ${\mathcal{L}}_{\alpha }^{\infty }$ arises as a scaling limit of random walks on discrete looptrees, and as a local limit of Brownian motion on compact stable looptrees, which has similar consequences for the limit of the heat kernel.

45 pages (some further proof details added to earlier version). arXiv admin note: text overlap with arXiv:1902.01713